Boundedness of solutions via the twist-theorem

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

R. D IECKERHOFF

E. Z EHNDER

Boundedness of solutions via the twist-theorem

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 14, n

^o

1 (1987), p. 79-95

<http://www.numdam.org/item?id=ASNSP_1987_4_14_1_79_0>

© Scuola Normale Superiore, Pisa, 1987, tous droits réservés.

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http://www.numdam.org/

R. DIECKERHOFF - E. ZEHNDER

1. Introduction and results

a) -

^Results

It is well known that the

longtime

behaviour of a

timedependent

^nonlinear

differential

equation

f being periodic in t,

^{can be} very intricate. For

example,

^{there are}

equations having

unbounded solutions but with

infinitely

many ^zeroes and with

nearby

unbounded solutions

having randomly prescribed

^{numbers of zeroes and also}

periodic solutions,

^{see V.M. Alekseev}

[8].

^{K. Sitnikov}

[7]

and J. Moser

[4].

^In

contrast to such unboundedness

phenomena

^one may look for conditions on the

nonlinearity,

^{in addition to the}

condition,

^that

which allow to conclude that all the solutions of the

equation

^{are bounded. For}

example

every solution of the

equation

p(t + 1) = p(t) being continuous,

^{is bounded. This}

result, prompted by questions

of J.E. Littlewood in

[2],

^{is due to G.R. Morris}

[1].

^{Our aim is to extend the}

result to a more

general

but still very restricted class of

equations

^{for which in}

particular

^the

timedependence

^{is involved in the}

nonlinearity

^{in x.}

Pervenuto alla redazione il 9 Gennaio 1986 ed in forma definitiva il 31 Gennaio 1986.

(*)

Supported by

^the

Stiftung Volkswagenwerk.

THEOREM 1.

Every

^solution

x(t) of

^the

equation

with +

1)

⁼ and pj ^E

Coo,

^is

bounded,

^{i.e. it exists}

for

^{all t and}

We shall not make the bound

explicit

^{in terms of the} initial conditions

x(O)

and

£(0)

^of the solution. The

proof

^{of Theorem 1 is}

strictly

2-dimensional and based on J. Moser’s twist theorem

similarly

^{to the}

proof

of Morris’ result. The

underlying

^{idea is as follows.}

By

^{means of} transformation

theory

^the

equation

is outside of a

large

^{disc D in the} transformed into a Hamiltonian

equation having

^the

following property. Following

^{the solutions from} the section t ⁼ 0 to the section t ⁼ 1 defines a map, the time ¹

map ~

^{of the}

flow,

^which

is close to a so called twist map D.

By

^{means of} the twist-theorem

one finds

large

^{invariant curves}

diffeomorphic

^to circles and

surrounding

^the

origin

^{in the}

(x, x)-plane. Every

^{such curve} is the base of a time

periodic

^and

under the flow invariant

cylinder

^{in the}

phase

space

(~, x, t) x 3,

^which confines the solutions in its interior and which therefore leads to a bound of these solutions. It turns out that the solutions

starting

^{at t = 0 on} the invariant

curves are

quasiperiodic.

THEOREM 2. There is a

(large)

^w* &#x3E; 0 such

that for

every irrational number

w &#x3E; w*

satisfying

for

^all

integers

^{p and with} two constants

,Q

&#x3E; 0 and c &#x3E;

0,

^{there is a}

quasiperiodic

^solution

of (1.4) having frequencies (w, 1 );

^{i.e. there is a smooth}

function F(OI, 02) periodic of period ^1,

^{such that}

are solutions

of

^the

equation.

It is well known that the invariant curves

guaranteed by

the twist-theorem lead to an abundance of

periodic

solutions. For

example by applying

^the

Poincare-Birkoff fixed

point

^{theorem to the} annuli bounded

by

^{two suitable}

invariant curves one finds fixed

points

^and

periodic points

of the time 1 map of the flow.

They give

^{rise to} forced oscillations and subharmonic solutions of the

equation.

^One

^knows,

ⁱⁿ

^addition,

that these invariant curves are in the closure of the set of

periodic points.

^{Since in our case the set} of invariant curves has infinite

Lebesgue

^measure

in J? 2 B D

^we shall conclude:

THEOREM 3. _{For every}

integer

m &#x3E; 1 there are

infinitely

many

periodic

solution

of (1.4) having

^minimal

period

^{m. Moreover} the closure

of

^{the set}

of

subharmonic solutions

of (1.4)

^is

of infinite Lebesgue

^{measure in the}

phase

x

S’ 1.

For related results

concerning infinitely

many forced oscillations ^we

point

out

[5]

^and

[6].

^{We should}

mention,

that up ^{to now} there is no

genuine generalization

^{of the first}

part

^{of this} statement to

higher dimensions,

^i.e.

equations

^{x =}

JV h(t, x),

^x

E ’~2n,

n &#x3E; 2. In the

special

^{case of}

equations

^of

the form i ⁼

JVh(x)

⁺

p(t) Berestycki

^{and Bahri}

[13]

^found

infinitely

many forced oscillations

using

^minimax

techniques,

^{under suitable}

assumptions

^{on the}

Hamilton function h.

b) -

^{Remark about} the smoothness

requirements.

It is

likely

that in the

general

^case

(1.1)

the boundedness of the solution is related to an excessive smoothness

requirement

in the x-variable. The role of the smoothness in the t-variable is not clear. In the

special

^{case of}

(1.4)

^where

the time is not involved in the nonlinear term, the

dependence

^{on t is}

only required

^{to be} continuous. In fact we shall prove:

THEOREM 4.

Every

^solution

of

with a continuous

p(t + 1)

⁼

p(t)

is bounded. Moreover the statements

of

^Theorem

2 and Theorem 3 hold true.

In this statement,

conjectured already

ⁱⁿ

[ 1 ],

^the

timedependent

^{term is}

bounded and is

required merely

^{to be} continuous. In contrast, the

proof

^{of the}

general

^case in Theorem 1 with the

timedependent

^term

unbounded, requires

an excessive amount of derivatives in the t-variable. In fact the number of derivatives in t we need

depends

^{on the size of the}

t-dependent

^{nonlinear term.}

It will follow from the

proof

^{that the statement of} Theorem 1 holds true for

the

equation

^°

0 .~

2n,

under the

following

smoothness

assumption

^{which is weaker than}

in Theorem _{1: pj} E where v ⁼

vel, n)

^{is the smallest}

integer satisfying

It is not clear whether the boundedness

phenomenon

is related to the smoothness in the t-variable or whether this

requirement

^{is a}

shortcoming

^{of our}

proof.

We

point

^{out that related}

problems

^{for the}

equation x

^{+ = 0 have been}

investigated by

^{Coffmann and Ullrich}

^{[ 17].}

^{We should} remark that the statement of Theorem 1 holds true for a more

general

^{class of}

equations

^{as can be seen}

from its

proof.

2. - Action and

angle-variables

Dropping

^the

timedependent

^term

equation (1.4)

^{becomes x +}

^{x2n+l =}

^0.

Introducing

^{x =} y ^we have vectorfield _{x = y}

and ^-x2n+’

^{which is a time-}

dependent

Hamiltonian

system

^on

Clearly h

&#x3E; 0 on

R2 _except

at the

only equilibrium point (x, y) = (o, 0)

^where

h ⁼ 0. All the solutions of

(2.1 )

^are

periodic,

^the

periods tending

^{to zero as h = E}

tends to

infinity.

^In

fact,

^{since the} Hamiltonian function is

homogeneous

^{in the x-}

variable,

the solutions are

easily

^{described in terms of a}

single

^{reference solution}

for which we take

(x*(t), y*(t)) having

the initial conditions

(~* (o), y*(O) = (1, 0).

It has _{the energy} , Let T* &#x3E; 0 be its minimal

period

and introduce the functions C and ~S’

by

’

These

analytic

^functions

satisfy

Let r &#x3E; 0 and _{set 1}

:- 1 ,

then the solution of

~ 2.1 )

with the initial conditions

n

(x(O), y(0)) = (r~’, 0)

^is

given by (x(t), y(t)) =

^{as one}

readily

verifies. It has

period

^{T In terms of its} energy ^we

find for T ⁼

f (E)

^{the function}

f (E)

⁼ C.E-11 with a ⁼

n . Loosly speaking

2(n + 1 )

the solutions

spin

^{around faster and faster with}

increasing

energy, which

has,

as it turns out, ^a

stabilizing

^effect.

The action and

angle

^{variables are now defined}

by

^{the )}

R 2 B {0},

^where

(x, y) = 0)

^{with A} &#x3E; 0 and with 0

(mod 1)

^is

given by

^the

formulae:

We claim

that 0

^{is a}

symplectic diffeomorphism

^{from R+ x}

^S

^{1 onto}

~~ 2 B 101-

Indeed,

for the Jacobian A

of 0

^{one finds}

by (ii)

^and

(iii) ^1,

^{so that}

1/J

^{is measure}

preserving.

Moreover since

(S, C)

^{is a solution of a} differential

equation having

^{T* as minimal}

period

^{one concludes}

that 0

^{is one to one and}

onto, which proves the claim.

In the new coordinates the Hamiltonian function

(2.1)

^becomes

which is

independent

^{of the}

angle

variable 0 so that the

system (2.1 )

becomes very

simple:

The full

equation (1.4)

has the Hamiltonian function:

Under the

symplectic transformation y

^{it is} transformed into:

where G is of the form:

with

Gj

^E

COO(T2)

^and

^T2

⁼

L~ 2/22.

The Hamiltonian

system Xh,,

^{is now more}

complicated:

In the

following

^we shall transform this

system,

^for

large A,

^{into a}

simpler

system

^{for which the}

0-depending

^{terms are}

^small,

^so that in the new variables A is close to an

integral.

To this effect an iterative _sequence of

finitely

many canonical transformations of R+ x

S’

¹ which

depend periodically

^on time t will be carried out in the next section.

3. - More canonical transformations

First we introduce a space of functions

fer)

which behave for

large A

&#x3E;

0,

with all their

derivatives,

^like

Arf,(O,t).

^{Given r E R we denote}

by fer)

^the

set of C°° functions in

(A, 0, t)

^x

T~

which are defined in A &#x3E;

Ao

^{for some}

Ao

&#x3E; 0 and for which there is a sequence

Àjlk

&#x3E; 0 such that

We summarize some

properties readily

verified from the definition:

LEMMA 1.

For

f

^E

T(r)

^we denote the meanvalue over the 0-variable

by [ f ] :

If

Ao

&#x3E;

0,

then

Aao

^{C R+ x}

^T2

denotes the annulus

PROPOSITION 1. Let

with

h,

E

fee)

^and

h2

^E

f(b).

^{Assume a} &#x3E;

1,

^{b a and c a.} Then there is ^a canonical

diffeomorphism 0 depending periodically

^{on t}

of

^the

form

with u c

~(1 - (a - b))

^{and v E}

.7(-(a - b))

^{such that}

A,,,

^{C C}

A~,_ for

some

large

J-Lo Moreover the

transformed

Hamiltonian

vectorfield

~b* (XH)

⁼

XH

^is

of

^the

form:

where

hi

^E

F(cl),

^with cl ⁼

max{c, bl,

^is

given by

and where

h2

^E

1(bl).

^{The constant}

bl

^{is smaller} that b and is

given by

PROOF. The

proof

will follow from several Lemmata. We shall look for the

requires reansfonnation 1b given by

^{means of a}

generating

^function

0, t),

so that

1/Jis implicitely

^defined

by

In the

following

^we shall often supress the variable t in the formulae.

Abbreviating:

we have for the transformed Hamiltonian vectorfield ⁼

fl expressed

in the variables

(ti, 0)

^{instead of}

(IL, 0):

By Taylor’s

^{formula we can write}

with

where ’ stands for the derivative in A. We now determine v from the

equation

so that

and therefore

Consequently:

LEMMA 2. Let S be

defined by (3.7)

^and

(3.6).

^{Then the}

formula (3.1) defines

a

symplectic diffeomorphism 0 depending periodically

^{on time t}

of

^the

form

PROOF. We first claim that

Indeed,

^{since a} &#x3E; c we

find,

^for 1L

sufficiently large,

and therefore

by

^Lemma

Then in view of Lemma 1

(iii)

91 - g2 ^{= v E}

~( 1 - (a - b))

and the claim

(3.9)

^{follows. Next we solve the}

second

equation

^of

(3.1)

^{for 0 and write}

For the inverse we set

In view of

(3.10)

^{we find} for v the

equation

If p

^is

large, then

^{so that v is}

uniquely

determined

by

^the contraction

principle. ^Moreover, by

^the

implicit

^{function theorem v E for some}

large

po. We claim

Indeed, apply

^{to the}

equation (3.11).

^The

right

^{hand side is a sum of}

terms

s

with 1 s + k k. The

highest

^{order term is the one}

~ k=1

with s = 0 and k ⁼

0, namely

^{which we}

put

^to the left hand side of the

equation. Inductively assuming

^that

for j

^{n - 1} the estimates

I

^{hold true we} conclude the same estimate

for j =

n, since g E

3~(-(a - b))

^{and therefore The claim}

(3.12)

^follows.

We next insert 0

= Q + v

into the first

equation

^of

^{(3.1 )}

^{and define u to be}

Since v E

7’(1 - (a - b))

^{in view of}

(3.9)

and since v E

~(- (a - b))

in view of

(3.12)

^{one concludes}

using (3.13)

^{that u E}

~( 1 - (a - b)).

To finish the

proof

of the Lemma one verifies

easily

^{that the}

map y

^{has a}

right

inverse of the same

type as 0

^{defined on}

A~_

^{for some}

large

1L- and that it is

injective

^{on for} po

large

For the transformed Hamiltonian function

ÎI,

^now

expressed

in the variables

J-L, cP

^{we have in view of}

(3.8), (3.5)

^and

(3.14):

where R ⁼

R,

⁺

R2

⁺

R3,

^with

LEMMA 3.

PROOF.

ad(i)

^{,S E}

~(1 - (a - b)) by (3.9)

^{and v E}

.1( -(a - b)) by

^Lemma

2,

hence

Rl

^E

~( 1 - (a - b)).

ad(ii)

^Set

f

^{= then}

f

^E

7(a - 2). Setting w

:= ~ ⁺ Tu, D ⁼

D,,

^then

is a sum of terms:

s

with 1 n amd 1 s n. Since

f

^E

F(a - 2)

^and

k=l

u E

~( 1 - (a - b))

^we

have ^Moreover, if j =

^{1 then}

2

for 1L sufficiently large,

^{since Du E}

~(2013(a - b)). If j

&#x3E;

1,

^then

Djw

⁼

TDJU

E

~(1 2013 (a - b) - j).

Therefore the above term is estimated

by J-La-2-n.

^Hence gi

:= f (w)

^E

F(a - 2).

^Since g2 .

u2

E

~(2 - 2(a - b))

^we

conclude in view of Lemma 1

(iii)

^that 91 - 92 C

F(b - (a - b))

^{as claimed.}

ad(iii)

Recall that

Dn( f (A, B))

^{is a sum of terms}

with and and Since

and one concludes that

Therefore as claimed.

In view of Lemma 2 and Lemma 3 and in view of

(3.15)

^the

proof

^of

Proposition

¹ is finished

setting h2

^{= R.}

4. - Proof of the Theorems 1-3

If H satisfies the

assumptions

^of

Proposition

¹ then for any

given

^number

d 0 there is an

integer j = j(a, b, d)

^{so that}

after j

successive

applications

^of

the

proposition

^we find that the

corresponding perturbation

^term

h2 belongs

^to

with bj

^d. We shall make the

number j

^of

steps

^needed

precise

^{for our}

special

^case

(1.7),

^{which in}

angle

^and action variables is a Hamiltonian

system

of the form:

where - - and

a = 1

^{* Therefore 1 a 2 and b a so} that the n+2

assumptions

^of

Proposition

^{1 are met.}

PROPOSITION 2. Let H be as in

(4.1 ).

^{Then there is a canonical}

transformation, periodic

^{in time t :}

V) =

^with

A,,,

^{C C}

A,,- for

^some IL- u+, which

transforms

the Hamiltonian

system

^into

~*(XH)

⁼

HH,

^where

with

hl

E

Y(b)

^and

h2

^E

~(-E) for -

&#x3E; 2 - a. The

number j of transformations

is smaller or

equal to j* = j * (n, t) where j*

^{is the smallest}

integer

&#x3E;

n (.~

⁺

³⁾

in the case that f n +

1, respectively

^{in the case}

PROOF. +

1,

^{so that b =}

a(.~ + 1)

^1.

Then j applications

^of

Proposition

^{1 lead to a}

perturbation

^term

h2 E 7’(bj) with bj =b-j(a- ^{1) =}

a(£ + 1) - jan,

^{which is} smaller than -2a

&#x3E; ~(£ + 3).

On the other

n

hand,

&#x3E; 1. Set £ + 1 ⁼ 2n + 2 - s, then after r

applications

^of

Proposition

^{1 we}

have br

⁼

a(2n

⁺

2) -

^{2r sa as}

long

^as

br &#x3E;

^1.

Now br

&#x3E; 1 if and

only

^{if 2rS n. Let k be so that}

^2ks

ⁿ

^2k+1s.

^Then

bk+l

^{= 1 and therefore}

bk+l+q

⁼

bk+l -qan,

^hence

bk+l+q

^-2a

hence in

particular

^{if We find in this case}

hence the result,..

Let now H be as in

(4.2)

^i.e.

on for some

large Ao,

^with

h,

^E

T(b)

^and

h2

^E

T( -6)

^{for 6} &#x3E; 2 - a. The

Hamiltonian equations

^are

Since

h2

^E

~(-~)

^{one verifies}

easily

that the solutions do exist for 0 t

1,

if the initial value

A(O) = A

^is

sufficiently large.

^{In fact we shall}

conclude

that the time 1 map ^is close to a twist map.

’

LEMMA 4. The time 1 map

ø1 of

^the

flow Ot of

^the

vectorfield XH given by (4.3)

^is

of

^the

form:

with Moreover

for

every

pair (r, s):

PROOF. Set

and set for the flow

(A(t), 0(t)) = 0)

^with

00

^=id

Then the

integral-equation

for the flow is

equivalent

^{to the}

following equations

^{for A and B:}

One verifies

easily

^{that for A} &#x3E;

Ao

^these

equations

^{have a}

unique

^{solution in}

the

space IAI,

¹

using

the contraction

principle,

^moreover A and B are smooth. The

required

^{estimates can then}

inductively

^be verified from

(4.6)

ⁱⁿ

view of

l(b),

^hence

aÀ2hl E a2 ^1(b-2),

^and

^{h2 E ~’(- ~). ·}

For the

completness

^{we include a}

proof

^{of the}

following

well known fact

(see

^f.e.

[14]).

LEMMA 5. The

map 7 = ~~

has the intersection property ^on

AÀo’

^i.e.

if

C is ^an embedded circle in

AÀo homotopic

^{to a circle A =const in}

AÀo’

^then

PROOF.

Since 7 = 01

^{1 is the time 1} map of ^{an exact} Hamiltonian

vectorfield, XH

^{it is exact}

symplectic,

^i.e.

^l*w-w

⁼ dW for some W on

AAO,

^{where w = Ad0.}

In

fact,

^since

ØO

^=id:

where ,

Therefore,

^{if ; 1 is the}

injection

map, ^we have and hence

Assume now,

by contradiction,

^that

7(C)

^{n C =}

0,

^then

7(C)

^{U C bounds an}

annulus in which has

positive

^measure.

Consequently, by

^Stokes

^formula,

in contradiction to

(4.7).

This proves the Lemma

Finally,

^{in the new} coordinates

the time 1

map ?

^is

expressed

^{as follows:}

From Lemma 4 one concludes the

following

^{estimates for the}

perturbation:

if _IL &#x3E;

s). Therefore,

^if it is

sufficiently large,

^the

map f ^is,

^{with its}

derivatives,

^{close to a standard twist} map.

Moreover,

it has the intersection

property

in view of Lemma

5,

^{so that the}

assumptions

^{of the} twist-theorem

[3]

are met.

It follows that for w &#x3E;

w*,

W*

sufficiently large,

^and

for two constants

,C3

&#x3E; 0 and c &#x3E; 0 and for all

integers

p and

q fl

^{0 there is}

an

embedding 0 : ^{S 1 -+}

^{of a}

circle,

^{which is}

differentiably

^{close to the}

injection map j

of the circle

(w)

^X

^{,S 1 --~}

and which is invariant under the map

,~. Moreover,

^on this invariant curve the

map ~

^is

conjugated

^{to a rotation}

with rotation number w :

The solutions of the Hamitonian

equation starting

^{at time t = 0 on} this invariant

curve determine a

1-periodic cylinder

^{in the} space

(p, 0, t)

^{E x ~ . Since}

the Hamiltonian vectorfield

XH

^is

timeperiodic,

^the

phase

space is ^x

S’ .

Let

(DI

with =id be the flow of the

time-independent

vectorfield

(XH,1 )

^on

Alto

^x

^S’

^{1 and} define the embedded torus T :

T 2 ~

x

S’

¹

by setting

In view of

(4.12)

^{we have}

with # = $ ’

^indeed

‘~’(s + 1, T) _ ~’( 1, T + 1 ) _ ~’(s, T).

Moreover ^o

T)

⁺

wt,

^{T +}

t),

^{so that the torus}

~’(T2)

^is

quasiperiodic

having

^the

frequencies (w, 1).

This proves ^{the statement} of Theorem 2. In order to prove the ^statement of Theorem 1

just

^observed

^that,

^{in the}

original coordinates,

every

point (x, y)

^E

^{R 2}

is in the interior of some invariant curve of the time 1 map of the flow which goes around the

origin.

Its solution is therefore confined in the interior of the time

periodic cylinder

^{above the invariant curve and hence}

is bounded. This ends the

proof

of Theorem 1. ·

We next prove Theorem 3.

Following

^at first the

arguments

of G. Morris

[1]

^we shall establish fixed

points

^of the iterated map

-~m

for every

integer

m

applying

^the Poincare-Birkoff fixed

point

^theorem

[9]

^to the map

#’~

in the annulus R bounded

by

^{two invariant curves} with rotation numbers

satisfying (4.11).

^{First we} map this annulus R onto an annulus

Ro = {(ç,1])IO ~ 1, ~

^mod

1}

^bounded

by

concentric circles.

If

wj

^{are the}

embeddings

^of the invariant curves

(see (4.12)) having

^rotation

numbers wj

^{we define} the map X : R

by

Since the

embeddings 1/;j

^are

differentiably

^{close to the}

injections

^{of the circles}

wj =const, x is a

diffeomorphism.

^{The induced} map G :

X- ^{I o Fo}

x,

expressed by

satisfies in view of

(4.12) f (~ + 1, r~)

⁼

f (~, n)

^and

g(~ +

⁼

g(~, 1]).

^{It leaves}

the boundaries _q ⁼ 0 and _q ⁼ 1 invariant and satisfies on these boundaries:

Define _{the map} s :

Ro - Ro by

⁼

(~

⁺

1,7y).

^{To find a}

periodic point

^of

minimal

period q &#x3E;

^{1 of}

G,

for any

given integer

q, ^we let

[qw]

^{be the}

integral

part

^of qw, i.e. _{qw - 1}

[qw]

qw, and choose an

integer

⁰

p

q such that

p + [qw]

^and q ^are relative

prime.

The map

Gq :

^on

^Ro, expressed by

satisfies the

required

^{twist condition at} the boundaries. Indeed with 0 a ⁼ qw -

[qw]

^{1 we} conclude from

(4.16):

Since

Gq

^{leaves a}

^regular

^{measure invariant we conclude that}

Gq

^{possesses a}

fixed

point

ⁱⁿ

Ro.

One verifies

readily

^{that it}

corresponds

^{to a}

periodic point

of

G,

^hence of

t,

with minimal

period

^{q. This} proves the first statement of Theorem 3.

As for the second statement of Theorem 3 we observe that the invariant

curves found

by

J. Moser’s twist-theorem are in the closure of the set of

periodic points.

^{This is well known and} follows from an

application

^{of the} Birkoff-Lewis fixed

point theorem,

^see e.g.

[15].

^{On the} other hand it is also well

known,

^that the set E c A covered

by

^{the invariant curves contained in an} annulus A fill out a set of

relatively large

^measure

m(E) &#x3E; (1 - E) - m(A)

^{with 0 E}

1,

^we refer to J. Pbschel

[16]

^{and the} references therein.

Applying

these observations to the annuli

A : k p k + I

^{for all}

large integers

^{one sees that the set of}

subharmonic solutions of

(1.4)

^{is indeed of infinite}

Lebesgue

^{measure, since}

This proves the ^statement of Theorem 3. ·

5. - Proof of Theorem 4

In case £ ⁼ 0 the

equation (1.7), expressed

in action and

angle-variables (a, 8), (2.3) becomes,

after the transformation ⁼ Àa with a

= -20132013:

n+2

and

fj being

continuous in t. One verifies

readily

that the time 1 flow of this

equation

^satisfies

with gj being analytic

^{and :S}

J-L-ê

^for p

&#x3E; p*(r, s).

^Therefore the

assumptions

^{of the} twist-theorem are met and Theorem 4 follows

readily arguing

^{as at} the very end of the

proof

of Theorem 1.

As for the Remark

(1.8)

^{we observe that in}

Proposition

^{2 one loses one}

derivative in t at every successive iteration

step,

which is due to the derivative in the

term a s _occurring

in the transformation formula. This

explains

^our

(9t .

^{g P}

smoothness

requirement.

As for the smoothness in the x variable we should remark that the twist-theorem

requires merely C"16-small perturbations

^{for those}

rotation numbers which we have considered above

[11],

^{and even}

^C3-small perturbations

for certain other irrational rotation numbers

[12].

The work has been

supported by

^the

Stiftung Volkswagenwerk.

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[2]

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